- FindStat

Identifier

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000337: Permutations ⟶ ℤ

Values

[.,.] => [1] => [] => [] => 0

[.,[.,.]] => [2,1] => [1] => [1] => 0

[[.,.],.] => [1,2] => [1] => [1] => 0

[.,[.,[.,.]]] => [3,2,1] => [2,1] => [1,2] => 0

[.,[[.,.],.]] => [2,3,1] => [2,1] => [1,2] => 0

[[.,.],[.,.]] => [3,1,2] => [1,2] => [2,1] => 1

[[.,[.,.]],.] => [2,1,3] => [2,1] => [1,2] => 0

[[[.,.],.],.] => [1,2,3] => [1,2] => [2,1] => 1

[.,[.,[.,[.,.]]]] => [4,3,2,1] => [3,2,1] => [1,2,3] => 0

[.,[.,[[.,.],.]]] => [3,4,2,1] => [3,2,1] => [1,2,3] => 0

[.,[[.,.],[.,.]]] => [4,2,3,1] => [2,3,1] => [2,1,3] => 1

[.,[[.,[.,.]],.]] => [3,2,4,1] => [3,2,1] => [1,2,3] => 0

[.,[[[.,.],.],.]] => [2,3,4,1] => [2,3,1] => [2,1,3] => 1

[[.,.],[.,[.,.]]] => [4,3,1,2] => [3,1,2] => [1,3,2] => 1

[[.,.],[[.,.],.]] => [3,4,1,2] => [3,1,2] => [1,3,2] => 1

[[.,[.,.]],[.,.]] => [4,2,1,3] => [2,1,3] => [2,3,1] => 1

[[[.,.],.],[.,.]] => [4,1,2,3] => [1,2,3] => [3,2,1] => 1

[[.,[.,[.,.]]],.] => [3,2,1,4] => [3,2,1] => [1,2,3] => 0

[[.,[[.,.],.]],.] => [2,3,1,4] => [2,3,1] => [2,1,3] => 1

[[[.,.],[.,.]],.] => [3,1,2,4] => [3,1,2] => [1,3,2] => 1

[[[.,[.,.]],.],.] => [2,1,3,4] => [2,1,3] => [2,3,1] => 1

[[[[.,.],.],.],.] => [1,2,3,4] => [1,2,3] => [3,2,1] => 1

[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0

[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0

[.,[.,[[.,.],[.,.]]]] => [5,3,4,2,1] => [3,4,2,1] => [2,1,3,4] => 1

[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [4,3,2,1] => [1,2,3,4] => 0

[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [3,4,2,1] => [2,1,3,4] => 1

[.,[[.,.],[.,[.,.]]]] => [5,4,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1

[.,[[.,.],[[.,.],.]]] => [4,5,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1

[.,[[.,[.,.]],[.,.]]] => [5,3,2,4,1] => [3,2,4,1] => [2,3,1,4] => 1

[.,[[[.,.],.],[.,.]]] => [5,2,3,4,1] => [2,3,4,1] => [3,2,1,4] => 1

[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [4,3,2,1] => [1,2,3,4] => 0

[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [3,4,2,1] => [2,1,3,4] => 1

[.,[[[.,.],[.,.]],.]] => [4,2,3,5,1] => [4,2,3,1] => [1,3,2,4] => 1

[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [3,2,4,1] => [2,3,1,4] => 1

[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [2,3,4,1] => [3,2,1,4] => 1

[[.,.],[.,[.,[.,.]]]] => [5,4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 1

[[.,.],[.,[[.,.],.]]] => [4,5,3,1,2] => [4,3,1,2] => [1,2,4,3] => 1

[[.,.],[[.,.],[.,.]]] => [5,3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 2

[[.,.],[[.,[.,.]],.]] => [4,3,5,1,2] => [4,3,1,2] => [1,2,4,3] => 1

[[.,.],[[[.,.],.],.]] => [3,4,5,1,2] => [3,4,1,2] => [2,1,4,3] => 2

[[.,[.,.]],[.,[.,.]]] => [5,4,2,1,3] => [4,2,1,3] => [1,3,4,2] => 1

[[.,[.,.]],[[.,.],.]] => [4,5,2,1,3] => [4,2,1,3] => [1,3,4,2] => 1

[[[.,.],.],[.,[.,.]]] => [5,4,1,2,3] => [4,1,2,3] => [1,4,3,2] => 1

[[[.,.],.],[[.,.],.]] => [4,5,1,2,3] => [4,1,2,3] => [1,4,3,2] => 1

[[.,[.,[.,.]]],[.,.]] => [5,3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 1

[[.,[[.,.],.]],[.,.]] => [5,2,3,1,4] => [2,3,1,4] => [3,2,4,1] => 2

[[[.,.],[.,.]],[.,.]] => [5,3,1,2,4] => [3,1,2,4] => [2,4,3,1] => 1

[[[.,[.,.]],.],[.,.]] => [5,2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 1

[[[[.,.],.],.],[.,.]] => [5,1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 2

[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [4,3,2,1] => [1,2,3,4] => 0

[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [3,4,2,1] => [2,1,3,4] => 1

[[.,[[.,.],[.,.]]],.] => [4,2,3,1,5] => [4,2,3,1] => [1,3,2,4] => 1

[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [3,2,4,1] => [2,3,1,4] => 1

[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [2,3,4,1] => [3,2,1,4] => 1

[[[.,.],[.,[.,.]]],.] => [4,3,1,2,5] => [4,3,1,2] => [1,2,4,3] => 1

[[[.,.],[[.,.],.]],.] => [3,4,1,2,5] => [3,4,1,2] => [2,1,4,3] => 2

[[[.,[.,.]],[.,.]],.] => [4,2,1,3,5] => [4,2,1,3] => [1,3,4,2] => 1

[[[[.,.],.],[.,.]],.] => [4,1,2,3,5] => [4,1,2,3] => [1,4,3,2] => 1

[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [3,2,1,4] => [2,3,4,1] => 1

[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [2,3,1,4] => [3,2,4,1] => 2

[[[[.,.],[.,.]],.],.] => [3,1,2,4,5] => [3,1,2,4] => [2,4,3,1] => 1

[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [2,1,3,4] => [3,4,2,1] => 1

[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [1,2,3,4] => [4,3,2,1] => 2

[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0

[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0

[.,[.,[.,[[.,.],[.,.]]]]] => [6,4,5,3,2,1] => [4,5,3,2,1] => [2,1,3,4,5] => 1

[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0

[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [4,5,3,2,1] => [2,1,3,4,5] => 1

[.,[.,[[.,.],[.,[.,.]]]]] => [6,5,3,4,2,1] => [5,3,4,2,1] => [1,3,2,4,5] => 1

[.,[.,[[.,.],[[.,.],.]]]] => [5,6,3,4,2,1] => [5,3,4,2,1] => [1,3,2,4,5] => 1

[.,[.,[[.,[.,.]],[.,.]]]] => [6,4,3,5,2,1] => [4,3,5,2,1] => [2,3,1,4,5] => 1

[.,[.,[[[.,.],.],[.,.]]]] => [6,3,4,5,2,1] => [3,4,5,2,1] => [3,2,1,4,5] => 1

[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0

[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [4,5,3,2,1] => [2,1,3,4,5] => 1

[.,[.,[[[.,.],[.,.]],.]]] => [5,3,4,6,2,1] => [5,3,4,2,1] => [1,3,2,4,5] => 1

[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => [4,3,5,2,1] => [2,3,1,4,5] => 1

[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => [3,4,5,2,1] => [3,2,1,4,5] => 1

[.,[[.,.],[.,[.,[.,.]]]]] => [6,5,4,2,3,1] => [5,4,2,3,1] => [1,2,4,3,5] => 1

[.,[[.,.],[.,[[.,.],.]]]] => [5,6,4,2,3,1] => [5,4,2,3,1] => [1,2,4,3,5] => 1

[.,[[.,.],[[.,.],[.,.]]]] => [6,4,5,2,3,1] => [4,5,2,3,1] => [2,1,4,3,5] => 2

[.,[[.,.],[[.,[.,.]],.]]] => [5,4,6,2,3,1] => [5,4,2,3,1] => [1,2,4,3,5] => 1

[.,[[.,.],[[[.,.],.],.]]] => [4,5,6,2,3,1] => [4,5,2,3,1] => [2,1,4,3,5] => 2

[.,[[.,[.,.]],[.,[.,.]]]] => [6,5,3,2,4,1] => [5,3,2,4,1] => [1,3,4,2,5] => 1

[.,[[.,[.,.]],[[.,.],.]]] => [5,6,3,2,4,1] => [5,3,2,4,1] => [1,3,4,2,5] => 1

[.,[[[.,.],.],[.,[.,.]]]] => [6,5,2,3,4,1] => [5,2,3,4,1] => [1,4,3,2,5] => 1

[.,[[[.,.],.],[[.,.],.]]] => [5,6,2,3,4,1] => [5,2,3,4,1] => [1,4,3,2,5] => 1

[.,[[.,[.,[.,.]]],[.,.]]] => [6,4,3,2,5,1] => [4,3,2,5,1] => [2,3,4,1,5] => 1

[.,[[.,[[.,.],.]],[.,.]]] => [6,3,4,2,5,1] => [3,4,2,5,1] => [3,2,4,1,5] => 2

[.,[[[.,.],[.,.]],[.,.]]] => [6,4,2,3,5,1] => [4,2,3,5,1] => [2,4,3,1,5] => 1

[.,[[[.,[.,.]],.],[.,.]]] => [6,3,2,4,5,1] => [3,2,4,5,1] => [3,4,2,1,5] => 1

[.,[[[[.,.],.],.],[.,.]]] => [6,2,3,4,5,1] => [2,3,4,5,1] => [4,3,2,1,5] => 2

[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0

[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [4,5,3,2,1] => [2,1,3,4,5] => 1

[.,[[.,[[.,.],[.,.]]],.]] => [5,3,4,2,6,1] => [5,3,4,2,1] => [1,3,2,4,5] => 1

[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => [4,3,5,2,1] => [2,3,1,4,5] => 1

[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [3,4,5,2,1] => [3,2,1,4,5] => 1

[.,[[[.,.],[.,[.,.]]],.]] => [5,4,2,3,6,1] => [5,4,2,3,1] => [1,2,4,3,5] => 1

[.,[[[.,.],[[.,.],.]],.]] => [4,5,2,3,6,1] => [4,5,2,3,1] => [2,1,4,3,5] => 2

[.,[[[.,[.,.]],[.,.]],.]] => [5,3,2,4,6,1] => [5,3,2,4,1] => [1,3,4,2,5] => 1

[.,[[[[.,.],.],[.,.]],.]] => [5,2,3,4,6,1] => [5,2,3,4,1] => [1,4,3,2,5] => 1

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 3 + 2\ q$

$F_{4} = 4 + 10\ q$

$F_{5} = 5 + 30\ q + 7\ q^{2}$

$F_{6} = 6 + 70\ q + 56\ q^{2}$

$F_{7} = 7 + 140\ q + 252\ q^{2} + 30\ q^{3}$

Description

The lec statistic, the sum of the inversion numbers of the hook factors of a permutation.
For a permutation

$\sigma = p \tau_{1} \tau_{2} \cdots \tau_{k}$ in its hook factorization, [1] defines

$\textrm{lec} \, \sigma = \sum_{1 \leq i \leq k} \textrm{inv} \, \tau_{i} \, ,$ where

$\textrm{inv} \, \tau_{i}$ is the number of inversions of

$\tau_{i}$ .

Map

restriction

Description

The permutation obtained by removing the largest letter.
This map is undefined for the empty permutation.

Map

to 132-avoiding permutation

Description

Return a 132-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the maximal element of the Sylvester class.

Map

complement

Description

Sents a permutation to its complement.
The complement of a permutation

$\sigma$ of length

$n$ is the permutation

$\tau$ with

$\tau(i) = n+1-\sigma(i)$